Enumeration of Complex Golay Pairs via Programmatic SAT

Abstract

We provide a complete enumeration of all complex Golay pairs of length up to 25, verifying that complex Golay pairs do not exist in lengths 23 and 25 but do exist in length 24. This independently verifies work done by F. Fiedler in 2013 [11] that confirms the 2002 conjecture of Craigen, Holzmann, and Kharaghani [8] that complex Golay pairs of length 23 don't exist. Our enumeration method relies on the recently proposed SAT+CAS paradigm of combining computer algebra systems with SAT solvers to take advantage of the advances made in the fields of symbolic computation and satisfiability checking. The enumeration proceeds in two stages: First, we use a fine-tuned computer program and functionality from computer algebra systems to construct a list containing all sequences which could appear as the first sequence in a complex Golay pair (up to equivalence). Second, we use a programmatic SAT solver to construct all sequences (if any) that pair off with the sequences constructed in the first stage to form a complex Golay pair.

Complex Golay pairs up to length 28: A search via computer algebra and programmatic SAT

“…Because Fiedler (2013), Gibson andJedwab (2011), andCraigen et al (2002) do not provide implementations or timings for the enumerations they completed it is not possible for us to compare the efficiency of our algorithm to previous algorithms. However, in Table 4 we compare our implementation's timings to the timings we previously presented (Bright et al, 2018b). Table 4 shows that the improved version of our algorithm performs about an order of magnitude faster in general.…”

“…A second possible improvement could be to use the symbolic form of f (θ) defined in Section 3.1 to help find the maximum of f (θ). For example, a consequence of Lemma 6 and Euler's identity is that Table 4: A comparison between the method presented at ISSAC 2018 (Bright et al, 2018b) and the method used in this paper. The times measure the total amount of computation time that was used to run a complete search in the lengths 17 ≤ n ≤ 28 when run on the same hardware.…”

“…One simple approach is to try a sufficiently large number of points z so that when a point exists with |A (z)| 2 > 2n (where A is either A even or A odd ) such a point is usually discovered. For example, Bright et al (2018b) tested Corollary 14 for 2 14 equally-spaced points z around the unit circle.…”

“…A preliminary version of our algorithm and results was presented at the conference ISSAC (Bright et al, 2018b). We have since made several improvements to our algorithm that allow us to extend our exhaustive search from all lengths n ≤ 25 to all lengths n ≤ 28.…”

Applying computer algebra systems with SAT solvers to the Williamson conjecture

“…Our focus was applying the SAT+CAS paradigm to the Williamson conjecture from combinatorial design theory, but we believe the SAT+CAS paradigm shows promise to be applicable to many other problems and conjectures. In fact, the SAT+CAS paradigm has recently been used to enumerate complex Golay pairs (Bright et al, 2018b) and good matrices (Bright et al, 2019). However, the SAT+CAS paradigm is not something that can be effortlessly applied to problems or expected to be effective on all types of problems.…”